Let a_i be the total number of aspirin consumed up to and including the i^th day, for i = 1, ..., 30. Combine these with the numbers a₁ + 14, ..., a₃₀ + 14, providing 60 numbers, all positive and less or equal 45 + 14 = 59. Hence two of these 60 numbers are identical. Since all a_i's and, hence, (a_i + 14)'s are distinct (at least one aspirin a day consumed), then a_j = a_i + 14, for some i<j. Thus, on days i + 1 to j, the person consumes exactly 14 aspirin.

In [Engel, p. 60] we find the following variant: A chessmaster has 77 days to prepare for a tournament. He wants to play at least one game per day, but no more than 132 games. Prove that there is a sequence of successive days on which he plays exactly 21 games.

References

1. A. Engel, Problem-Solving Strategies, Springer Verlag, 1998